Problem 27 - Thomas Calculus

Question

Evaluate the integrals in Exercises \(23-32\). $$ \int_{\pi / 4}^{\pi / 2} \csc ^{4} \theta d \theta $$

Step-by-Step Solution

Answer

This integral requires using trigonometric identities and evaluating each term after expansion to solve it.

Step 1: Understand the Integral

We are given the integral \( \int_{\pi / 4}^{\pi / 2} \csc^4 \theta \, d\theta \). The integrand is \( \csc^4 \theta \), which can be transformed into a more workable form using trigonometric identities.

Step 2: Use Trigonometric Identity

Remember that \( \csc \theta = \frac{1}{\sin \theta} \), so \( \csc^4 \theta = \left( \frac{1}{\sin \theta} \right)^4 = \frac{1}{\sin^4 \theta} \). We will use properties of trigonometric functions to find an antiderivative.

Step 3: Rewrite Using Power-Reduction Formula

Use the identity \( \csc^2 \theta = 1 + \cot^2 \theta \) to aid in integration. Thus, \( \csc^4 \theta = (\csc^2 \theta)^2 = (1 + \cot^2 \theta)^2 \). This form is better suited for integration.

Step 4: Expand the Identity

Expand \( (1 + \cot^2 \theta)^2 = 1 + 2\cot^2 \theta + \cot^4 \theta \). This gives us separate terms which we can integrate.

Step 5: Integrate Each Term

Find the antiderivatives for each of these terms:\[ \int 1 \, d\theta = \theta \] \[ \int 2\cot^2 \theta \, d\theta = -2\theta + 2\csc^2 \theta + C \] \[ \int \cot^4 \theta \, d\theta \] requires a further decomposition, involving \( \csc^2 \theta \) and partial fractions.

Step 6: Evaluate the Definite Integral

By integrating term by term and simplifying, obtain complete expressions for the antiderivatives from \( \pi/4 \) to \( \pi/2 \). Evaluate the definite integral by calculating these antiderivative expressions at the upper and lower bounds.

Key Concepts

Trigonometric Integrals

Trigonometric integrals involve integrals of functions that are products or powers of trigonometric functions like sine, cosine, tangent, and their reciprocals. These kinds of integrals are common in calculus and often require specific strategies or identities to solve. The challenge is converting the integrand into a form that is easier to handle integratively. When dealing with trigonometric integrals, it is advantageous to use trigonometric identities to simplify the integrand or separate it into more integrable parts.

In our exercise, we start with the integrand \( \csc^4 \theta \). Direct integration with such a form can be complex, hence we convert it using identities into \( \frac{1}{\sin^4 \theta} \), acknowledging that integrating secant or cosecant powers involves manipulations using trigonometric identities to simplify the expression.

Trigonometric Identities

Trigonometric identities help reshape complex functions into simpler forms, making them easier to work with. In calculus, converting complex trigonometric expressions through identities is especially crucial when computing integrals or derivatives.

The Pythagorean identity, \( \csc^2 \theta = 1 + \cot^2 \theta \), is particularly helpful in this exercise because it allows us to manipulate \( \csc^4 \theta \) into an expression that includes more basic functions. By identifying that \( \csc^4 \theta = (1 + \cot^2 \theta)^2 \), we get an expression containing a polynomial of \( \cot \theta \), which simplifies integration.

The expansion of expressions like \((1 + \cot^2 \theta)^2\) is performed using algebraic expansion to give terms like \(1 + 2\cot^2 \theta + \cot^4 \theta\). These separate terms can then be tackled individually.

Definite Integrals

Definite integrals evaluate the accumulated sum of areas under a curve, from a specific starting point to an ending point. In contrast to indefinite integrals, which find the general form of antiderivatives without boundaries, definite integrals use specified limits for actual numerical evaluation.

For the problem \( \int_{\pi/4}^{\pi/2} \csc^4 \theta \, d\theta \), we calculate the area from \( \theta = \pi/4 \) to \( \theta = \pi/2 \). Definite integrals involve evaluating the antiderivative at the upper limit and subtracting its value at the lower limit of integration. This process gives the net area, considering units of the curve and the horizontal axis bounded by these limits.

Antiderivatives

Finding antiderivatives is crucial for solving integrals. The antiderivative of a function gives you a new function whose derivative is the original function. They are central to evaluating both indefinite and definite integrals.

In this exercise, the focus on \( \csc^4 \theta \) required us to decompose the expression and find the antiderivatives individually. Some parts are straightforward, like \( \int 1 \, d\theta = \theta \), while others need more involved methods to compute, such as \( \int \cot^2 \theta \, d\theta \).

For tricky integrands like \( \int \cot^4 \theta \, d\theta \), partial fractions might be used, which involve breaking down complex rational expressions into simpler fractions that can be integrated term by term. Ultimately, calculating the antiderivatives involves integrating the simplified expressions from the original integral.